The profile of that tap looks like a Nut Tap. Long tapered lead in for production tap threading of nuts. Very low torque thread cutting because of that long lead in taper.
What a nut tap? ✅ When is it used? ✅ Nut taps are used to produce nuts with ISO standard metric threads ✅ Nut taps have a very long lead-in (2/3 of the thread length) ✅ are used for thread depths up to 1.5 x nominal diameter in short through holes ✅
threadingtoolsguide.com
1/32 might be the diameter change from full thread depth to the start of the thread on that tap. That looks much too coarse to be a thread pitch for that tap, assuming that it is 5/8 diameter. The pitch looks closer to .100" or 1/10.
Now for some clarity, according to a little research here, that is a Very Old tap.
Wells Brothers & Co. was incorporated in 1879.
That changed to Wells Brothers Co. in 1888.
That tap is older than 1888 and newer than 1879.
That very much pre-dates the Unified Thread system (UNC, UNF, UNS, etc.)
5/8-10 is no longer a standard thread in the Unified Standard. That size and tap was from the days of everyone making whatever they felt was correct for the application.
A quick measurement of the OD at the thickest portion and at the very end will probably say a lot.